Transcript Improving outcomes in trigonometry

Improving outcomes in
trigonometry
Understanding identities
Solving trigonometric problems
Strategies to engage students in effective
mathematical discussion
Teaching ideas that promote high level
problem solving
Students find it really hard
It’s very abstract
• I’ve deleted the joke
– It wasn’t funny
– Students don’t know who he is
– He’s not in favour these days
Understanding the identities
Reciprocal trig functions
– there’s one “co” in every pair!
1
 cosecx
sin x
1
 sec x
cos x
1
 cot x
tan x
Matching cards practise the definitions and the
basics of trig equations – the second value is the
tricky one!
One plus tan
is sexy
One plus coat
is cosy
1  tan 2   sec 2 
1  cot 2   cos ec 2
Proving these
from Pythagoras
is worth doing,
I think
Compound Angle Formulae
sin ( A + B ) = sin A cos B + cos A sin B
sin ( A - B ) = sin A cos B - cos A sin B
cos( A + B ) = cos A cos B - sin A sin B
cos( A - B ) = cos A cos B + sin A sin B
tan ( A + B ) = tan A + tan B
1 - tan A tan B
tan ( A - B ) = tan A - tan B
1 + tan A tan B
Proving this one
from the other two
is a good example
Double Angle Formulae
sin 2A = 2 sin A cos A
cos 2A = cos 2 A – sin 2 A
Using sin2 A + cos2 A = 1
cos 2A = 1 – 2 sin 2 A
there are 2 alternative forms for this equation:cos 2A = 2 cos 2 A – 1
tan 2A = 2 tan A
1 – tan 2 A
Learn these!
They are too important to need looking up each time!
Make sure if you forget, you can reconstruct them from the
ones in the formula book
Introduction to R alpha form
Investigative approach
This makes a good investigation using graphics
calculators or graph drawing software
Draw y  p cos x  q sin x
Use constant controller or sliders to change the
values of p and q – what do you notice?
How useful is
the technique?
Choose p = 2 and q = 3
Add y  R cos( x  a ) and find values of R and a
so that the two graphs coincide We could do with
an algebraic
method!
Expressing a cos θ + b sin θ in R alpha form
Express 2cos θ + 5 sin θ in the form R cos ( θ – α ) where R > 0 and 0 < α < 90°
2cos θ + 5 sin θ = R cos ( θ – α ) = R cos θ cos α + R sin θ sin α
R cos α = 2
R sin α = 5
dividing gives
tan α = 5/2
square and add gives
α = 68.2°
R2 = 22 + 52
R = √29
2cos θ + 5 sin θ =√29 cos ( θ – 68.2°)
Max value of 2cos θ + 5 sin θ =√29 when θ =68.2°
Min value of 2cos θ + 5 sin θ =-√29 when θ =180 +68.2°=248.2
If you found max and min using calculus,
could you get these exact values?
Hence solve for 0 < θ < 360°,
2cos θ + 5 sin θ = 3
√29 cos ( θ – 68.2°) = 3
cos ( θ – 68.2°) = 3/√29
the equation
2cos θ + 5 sin θ = 3
Choices go
in here – as
soon as you
use cos-1
θ – 68.2° = 56.1°
1 y
θ – 68.2° = -56.1°, 56.1°,303.9°
x
θ = 12.1°, 124.3°
90
56.1°
–1
180
270
360
Trig Parametric Equations
Show that the parametric equations
x  a  r cos  , y  b  r sin 
represent a circle and find the centre and radius
xa
y b
cos  
, sin  
r
r
cos 2   sin 2   1
 x a   y b

 
 1
 r   r 
2
2
( x  a ) 2  ( y  b) 2  r 2
which is the equation of circle centre (a,b) radius r
Problem solving
Using identities in proof
1  cos 2 A
Prove that
 tan A
sin 2 A
1  cos 2 A 1  (1  2sin A)

sin 2 A
2sin A cos A
2
Best strategy is to start
with one end and get
all the way to the other
Discussion here
Cos 2A is the one with
the choices to make!
2sin 2 A
sin A


 tan A
2sin A cos A
cos A
Discussion
point – what
do you do if
you run out of
steam?
High level problem solving
•
•
•
•
Don’t prepare the way
Don’t direct too early
Don’t block off the dead ends
Let them sweat over it a bit!!
• After they’ve had time to think, ask questions
• What topics might be useful
• Are there formulae you can use
• What would you need to know before you can answer the
question?
Plenary – nrich 1955
Plenary – nrich 1955
DC
tan(DQC ) 
1
QC
tan(DBC )  tan(DPC )
tan(DBC  DPC ) 
1   tan(DBC )  tan(DPC ) 
5
1 1

6
3
2

 1  tan(DQC )

1
 1  1 
1  
1    
5
 3  2 
DBC  DPC  DQC
Plenary nrich 1955
• According to the notes, this problem was done
in eight different ways by two A level students
– I’m guessing Further Maths students!
• See how at http://nrich.maths.org/1335/index
Any questions?
• Hope you’ve found the day valuable
• Safe journey home!